Of great importance to electromagnetic wave calculations is the WKB approximation. This is a high frequency approximation that leads to the ray tracing techniques which lie at the heart of many propagation calculations. Consider expression (6) for the propagation kernel, then the maximum contribution to the path integral will come from paths around that for which the phase distance S is minimum (the ray path of Fermat's principle). Around this path, the lowest order variation in S will be quadratic. Ignoring variations higher than quadratic, this yields the WKB approximation [Rosen, 1969]
where Sc and δ2Sc represent S and its second variation when evaluated on the path of minimum phase. (The remaining path integral is in terms of variations q from the path of minimum phase and for which q = 0 at the end points.) Consider the path of minimum phase to be parameterized by its group path τ (dτ = ) and denote the end points (x0, z0) and (x, z) by A and B, respectively. Let r = rc(τ) be the minimum phase path expressed in three- dimensional coordinates, then this path satisfies [Jones, 1979]
Consider path variations q of the form
where the vectors Pi are unit length, perpendicular to the minimum phase path (and to each other) and satisfy
Vectors Pi are the “polarisation” vectors of the standard WKB solution to Maxwells equations [Jones, 1979, p. 594]. The second variation δ2Sc can now be expressed in the form
where (x1, x2, x3) are Cartesian coordinates in three-dimensional space. The normalization kernel
can be built up from the short path kernels
where Δτ is small, matrix Γ has coefficients γij and subscripts A and B denote quantities evaluated at the beginning and end of the short path (note that the normalization in (17) ensures a delta function characteristic as Δτ 0). Consider the path of minimum phase that joins two general points and break this into N short paths, each with group path length Δτ. Repeated application of the short path kernel will yield
where the ith path runs between points qi−1 and qi ( note that q0 = qN = 0). The above integral can be evaluated to yield
where δij is the Kronecker delta. It will be noted that both C12 and F = C11 − C22 are diagonal matrices and that D = Δτ2∣C222 + C22F − C122∣. From the factorization
it follows that
Expanding the above determinant results in the difference equation
and, in the limit N ∞, the differential equation
with boundary conditions 1 and D± 0 as τ 0 (it has been assumed that n = 1 at the start of propagation).
 Equations (11), (13), (15) and (27) together provide a means of calculating the parameters that are required in the evaluation of the kernel
This kernel, together with integral equation (4), provides the required WKB solution to the Helmholtz equation (S(r, r0) = is the phase distance). Although the calculation of the kernel will require both P1 and P2 , it will be noted that P1, P2 and the minimum phase path are mutually orthogonal and so the solution of (13) will only be required for either P1 or P2. It will also be noted that a knowledge of P1 and P2 facilitates the calculation of a WKB solution to the full Maxwell equations for nonhomogeneous media (a fact will be exploited in the next section). At the beginning of propagation, the electric field will need to be expanded in the P1 and P2 directions. The evolution of these vectors can then be used to track the evolution of the electric field direction along a ray and the above WKB kernel its magnitude. This procedure will provide an effective approximate solution away from caustics and strong sources of diffraction.